Set (Mathematics)
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A set (also called a collection; English: set) is a grouping of objects that are clearly distinguished from one another. The objects contained in a set are called elements. Membership is fundamental: any given object either is or is not an element of the set; there is no middle ground. Usually, the number of times an element appears is not counted, and the order of elements does not matter.
Sets are among the most basic concepts in modern mathematics. Numbers, functions, geometric shapes, and algebraic structures can all be defined starting from sets. This has led to set theory being used as a common language in which different branches of mathematics can communicate.
How to Define a Set
There are two common ways to describe a set. First, by listing the elements enclosed in braces ({}), for example {1, 2, 3}. Second, by stating a condition that the elements satisfy, for example "all even positive integers". The second method is useful for infinite sets. A set with no elements is called the empty set (∅). Two sets are identical if and only if they have exactly the same elements.
Relations and Operations
Set A is a subset of set B if every element in A is also in B. Standard operations include union, which combines the elements of two sets; intersection, which takes elements that belong to both; and difference, which takes elements in one set but not the other. The complement is also defined when a larger universal set is specified.
Cardinality and Counting
The number of elements in a set is called its cardinality. Some sets are finite, such as {a, b, c} which has cardinality three. Others are infinite, such as the set of natural numbers. Georg Cantor, a German mathematician, developed in the nineteenth century methods for comparing the cardinalities of infinite sets using one-to-one correspondence. His results showed that not all infinite sets have the same cardinality.
Set Theory and Axioms
An unrestricted definition allowing sets to be formed from any condition leads to logical contradictions, such as Russell's paradox. For this reason, mathematicians constructed axiomatic systems—the most widely used being ZFC (Zermelo–Fraenkel with the axiom of choice)—that specify the permitted ways to construct new sets. These systems now form the standard foundation of mathematics.